1.Prove that the orthocentre of the triangle formed by the three tangents to a parabola lie on a directrix.

For the parabola , the tangent at the point has equation . Check that the tangents at the points and meet at the point . The line through this point perpendicular to the tangent at the point will have gradient , and its equation is therefore .

That line is one of the altitudes of the triangle formed by the tangents at the three points on the parabola. It meets the directrix at the point given by , or . That last equation is symmetric in p, q and r. So both the other altitudes of the triangle will pass through the same point on the directrix. In other words, the orthocentre of the triangle lies on the directrix.